In the context of type IIA string theory compactified on a Calabi-Yau (CY) threefold \(\mathfrak {Y}\) , BPS states correspond to objects in the bounded derived category of coherent sheaves \(\mathcal {C}=D^b\operatorname{Coh}(\mathfrak {Y})\) 
which are stable for a particular Bridgeland stability condition determined by the complexified Kähler
moduli, known as \(\Pi \) -stability [1]}, [2]}, [3]}, [4]}. For a general compact CY threefold, the construction of the space of Bridgeland stability conditions
\(\operatorname{Stab}(\mathcal {C})\)  is a difficult mathematical problem, and the identification of the submanifold \(\Pi \subset \operatorname{Stab}(\mathfrak {Y})\)  corresponding to \(\Pi \) -stability depends on the symplectic geometry of \(\mathfrak {Y}\)  (namely, its genus-zero Gromov-Witten invariants).
For fixed charge \(\gamma \in K(\mathfrak {Y})\)  and central chargeAs we recall in §REF ,
a stability condition
\(\sigma =(Z,\mathcal {A})\)  on \(\mathcal {C}\)  also involves a choice of Abelian subcategory \(\mathcal {A}\subset \mathcal {C}\)  (the heart).
We omit it here for brevity since it is locally determined by the central charge \(Z\) .
\(Z\) 
(determined by the Kähler moduli), stable objects are counted by the generalized
Donaldson-Thomas (DT) invariant \(\Omega _Z(\gamma )\)  [5]}, [6]}, [7]}. The latter, being integer valued, is locally constant but discontinuous across real-codimension one walls in \(\operatorname{Stab}(\mathfrak {Y})\)  (hence also on \(\Pi \) ), due to the (dis)appearance of destabilizing sub-objects, leading to an intricate chamber structure. While the jump is determined in terms of the invariants on one side of the wall by a universal wall-crossing formula [6]}, [7]}, it is desirable to develop a global understanding of the BPS spectrum which allows to identify stable objects at any point \(z\)  in \(\Pi \subset \operatorname{Stab}(\mathfrak {Y})\) .
